\(\int \frac {e^{-2 \arctan (a x)}}{(c+a^2 c x^2)^4} \, dx\) [308]

Optimal result

Integrand size = 21, antiderivative size = 124 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=-\frac {9 e^{-2 \arctan (a x)}}{160 a c^4}-\frac {e^{-2 \arctan (a x)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}-\frac {3 e^{-2 \arctan (a x)} (1-2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}-\frac {9 e^{-2 \arctan (a x)} (1-a x)}{80 a c^4 \left (1+a^2 x^2\right )} \] Output:

-9/160/a/c^4/exp(2*arctan(a*x))-1/20*(-3*a*x+1)/a/c^4/exp(2*arctan(a*x))/( 
a^2*x^2+1)^3-3/40*(-2*a*x+1)/a/c^4/exp(2*arctan(a*x))/(a^2*x^2+1)^2-9/80*( 
-a*x+1)/a/c^4/exp(2*arctan(a*x))/(a^2*x^2+1)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.48 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {8 c e^{-2 \arctan (a x)} (-1+3 a x)-3 \left (c+a^2 c x^2\right ) \left (e^{-2 \arctan (a x)} (4-8 a x)+3 (1-i a x)^{-i} (1+i a x)^i (-i+a x) (i+a x) \left (3-2 a x+a^2 x^2\right )\right )}{160 a c^2 \left (c+a^2 c x^2\right )^3} \] Input:

Integrate[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^4),x]
 

Output:

((8*c*(-1 + 3*a*x))/E^(2*ArcTan[a*x]) - 3*(c + a^2*c*x^2)*((4 - 8*a*x)/E^( 
2*ArcTan[a*x]) + (3*(1 + I*a*x)^I*(-I + a*x)*(I + a*x)*(3 - 2*a*x + a^2*x^ 
2))/(1 - I*a*x)^I))/(160*a*c^2*(c + a^2*c*x^2)^3)
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5593, 27, 5593, 5593, 5594}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{-2 \arctan (a x)}}{\left (a^2 c x^2+c\right )^4} \, dx\)

\(\Big \downarrow \) 5593

\(\displaystyle \frac {3 \int \frac {e^{-2 \arctan (a x)}}{c^3 \left (a^2 x^2+1\right )^3}dx}{4 c}-\frac {(1-3 a x) e^{-2 \arctan (a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \int \frac {e^{-2 \arctan (a x)}}{\left (a^2 x^2+1\right )^3}dx}{4 c^4}-\frac {(1-3 a x) e^{-2 \arctan (a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}\)

\(\Big \downarrow \) 5593

\(\displaystyle \frac {3 \left (\frac {3}{5} \int \frac {e^{-2 \arctan (a x)}}{\left (a^2 x^2+1\right )^2}dx-\frac {(1-2 a x) e^{-2 \arctan (a x)}}{10 a \left (a^2 x^2+1\right )^2}\right )}{4 c^4}-\frac {(1-3 a x) e^{-2 \arctan (a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}\)

\(\Big \downarrow \) 5593

\(\displaystyle \frac {3 \left (\frac {3}{5} \left (\frac {1}{4} \int \frac {e^{-2 \arctan (a x)}}{a^2 x^2+1}dx-\frac {(1-a x) e^{-2 \arctan (a x)}}{4 a \left (a^2 x^2+1\right )}\right )-\frac {(1-2 a x) e^{-2 \arctan (a x)}}{10 a \left (a^2 x^2+1\right )^2}\right )}{4 c^4}-\frac {(1-3 a x) e^{-2 \arctan (a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}\)

\(\Big \downarrow \) 5594

\(\displaystyle \frac {3 \left (\frac {3}{5} \left (-\frac {(1-a x) e^{-2 \arctan (a x)}}{4 a \left (a^2 x^2+1\right )}-\frac {e^{-2 \arctan (a x)}}{8 a}\right )-\frac {(1-2 a x) e^{-2 \arctan (a x)}}{10 a \left (a^2 x^2+1\right )^2}\right )}{4 c^4}-\frac {(1-3 a x) e^{-2 \arctan (a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}\)

Input:

Int[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^4),x]
 

Output:

-1/20*(1 - 3*a*x)/(a*c^4*E^(2*ArcTan[a*x])*(1 + a^2*x^2)^3) + (3*(-1/10*(1 
 - 2*a*x)/(a*E^(2*ArcTan[a*x])*(1 + a^2*x^2)^2) + (3*(-1/8*1/(a*E^(2*ArcTa 
n[a*x])) - (1 - a*x)/(4*a*E^(2*ArcTan[a*x])*(1 + a^2*x^2))))/5))/(4*c^4)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 5593

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> S 
imp[(n - 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)*(E^(n*ArcTan[a*x])/(a*c*(n^2 + 
4*(p + 1)^2))), x] + Simp[2*(p + 1)*((2*p + 3)/(c*(n^2 + 4*(p + 1)^2)))   I 
nt[(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] 
&& EqQ[d, a^2*c] && LtQ[p, -1] &&  !IntegerQ[I*n] && NeQ[n^2 + 4*(p + 1)^2, 
 0] && IntegerQ[2*p]
 

rule 5594

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E 
^(n*ArcTan[a*x])/(a*c*n), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c]
 

Maple [A] (verified)

Time = 47.54 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.60

method	result	size
gosper	\(-\frac {\left (9 x^{6} a^{6}-18 a^{5} x^{5}+45 a^{4} x^{4}-60 a^{3} x^{3}+75 a^{2} x^{2}-66 a x +47\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{160 \left (a^{2} x^{2}+1\right )^{3} c^{4} a}\)	\(75\)
parallelrisch	\(\frac {\left (-9 x^{6} a^{6}+18 a^{5} x^{5}-45 a^{4} x^{4}+60 a^{3} x^{3}-75 a^{2} x^{2}+66 a x -47\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{160 c^{4} \left (a^{2} x^{2}+1\right )^{3} a}\)	\(75\)
orering	\(-\frac {\left (9 x^{6} a^{6}-18 a^{5} x^{5}+45 a^{4} x^{4}-60 a^{3} x^{3}+75 a^{2} x^{2}-66 a x +47\right ) \left (a^{2} x^{2}+1\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{160 a \left (a^{2} c \,x^{2}+c \right )^{4}}\)	\(82\)

Input:

int(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/160*(9*a^6*x^6-18*a^5*x^5+45*a^4*x^4-60*a^3*x^3+75*a^2*x^2-66*a*x+47)/( 
a^2*x^2+1)^3/c^4/exp(2*arctan(a*x))/a
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=-\frac {{\left (9 \, a^{6} x^{6} - 18 \, a^{5} x^{5} + 45 \, a^{4} x^{4} - 60 \, a^{3} x^{3} + 75 \, a^{2} x^{2} - 66 \, a x + 47\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{160 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \] Input:

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="fricas")
 

Output:

-1/160*(9*a^6*x^6 - 18*a^5*x^5 + 45*a^4*x^4 - 60*a^3*x^3 + 75*a^2*x^2 - 66 
*a*x + 47)*e^(-2*arctan(a*x))/(a^7*c^4*x^6 + 3*a^5*c^4*x^4 + 3*a^3*c^4*x^2 
 + a*c^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\text {Timed out} \] Input:

integrate(1/exp(2*atan(a*x))/(a**2*c*x**2+c)**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\int { \frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}} \,d x } \] Input:

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="maxima")
 

Output:

integrate(e^(-2*arctan(a*x))/(a^2*c*x^2 + c)^4, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (111) = 222\).

Time = 0.15 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.26 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=-\frac {47 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{12} + 132 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{11} + 18 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{10} - 180 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{9} + 225 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} + 456 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{7} - 4 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} - 456 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{5} + 225 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} + 180 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{3} + 18 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} - 132 \, \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right ) + 47}{160 \, {\left (a c^{4} e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{12} + 6 \, a c^{4} e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{10} + 15 \, a c^{4} e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} + 20 \, a c^{4} e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} + 15 \, a c^{4} e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} + 6 \, a c^{4} e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} + a c^{4} e^{\left (2 \, \arctan \left (a x\right )\right )}\right )}} \] Input:

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="giac")
 

Output:

-1/160*(47*tan(1/2*arctan(a*x))^12 + 132*tan(1/2*arctan(a*x))^11 + 18*tan( 
1/2*arctan(a*x))^10 - 180*tan(1/2*arctan(a*x))^9 + 225*tan(1/2*arctan(a*x) 
)^8 + 456*tan(1/2*arctan(a*x))^7 - 4*tan(1/2*arctan(a*x))^6 - 456*tan(1/2* 
arctan(a*x))^5 + 225*tan(1/2*arctan(a*x))^4 + 180*tan(1/2*arctan(a*x))^3 + 
 18*tan(1/2*arctan(a*x))^2 - 132*tan(1/2*arctan(a*x)) + 47)/(a*c^4*e^(2*ar 
ctan(a*x))*tan(1/2*arctan(a*x))^12 + 6*a*c^4*e^(2*arctan(a*x))*tan(1/2*arc 
tan(a*x))^10 + 15*a*c^4*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^8 + 20*a*c^ 
4*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^6 + 15*a*c^4*e^(2*arctan(a*x))*ta 
n(1/2*arctan(a*x))^4 + 6*a*c^4*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^2 + 
a*c^4*e^(2*arctan(a*x)))
 

Mupad [B] (verification not implemented)

Time = 23.41 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {9\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (a\,x-1\right )}{80\,a\,c^4\,\left (a^2\,x^2+1\right )}-\frac {9\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}}{160\,a\,c^4}+\frac {3\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x-1\right )}{40\,a\,c^4\,{\left (a^2\,x^2+1\right )}^2}+\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (3\,a\,x-1\right )}{20\,a\,c^4\,{\left (a^2\,x^2+1\right )}^3} \] Input:

int(exp(-2*atan(a*x))/(c + a^2*c*x^2)^4,x)
 

Output:

(9*exp(-2*atan(a*x))*(a*x - 1))/(80*a*c^4*(a^2*x^2 + 1)) - (9*exp(-2*atan( 
a*x)))/(160*a*c^4) + (3*exp(-2*atan(a*x))*(2*a*x - 1))/(40*a*c^4*(a^2*x^2 
+ 1)^2) + (exp(-2*atan(a*x))*(3*a*x - 1))/(20*a*c^4*(a^2*x^2 + 1)^3)
 

Reduce [F]

\[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {-6 e^{2 \mathit {atan} \left (a x \right )} \left (\int \frac {x}{e^{2 \mathit {atan} \left (a x \right )} a^{8} x^{8}+4 e^{2 \mathit {atan} \left (a x \right )} a^{6} x^{6}+6 e^{2 \mathit {atan} \left (a x \right )} a^{4} x^{4}+4 e^{2 \mathit {atan} \left (a x \right )} a^{2} x^{2}+e^{2 \mathit {atan} \left (a x \right )}}d x \right ) a^{8} x^{6}-18 e^{2 \mathit {atan} \left (a x \right )} \left (\int \frac {x}{e^{2 \mathit {atan} \left (a x \right )} a^{8} x^{8}+4 e^{2 \mathit {atan} \left (a x \right )} a^{6} x^{6}+6 e^{2 \mathit {atan} \left (a x \right )} a^{4} x^{4}+4 e^{2 \mathit {atan} \left (a x \right )} a^{2} x^{2}+e^{2 \mathit {atan} \left (a x \right )}}d x \right ) a^{6} x^{4}-18 e^{2 \mathit {atan} \left (a x \right )} \left (\int \frac {x}{e^{2 \mathit {atan} \left (a x \right )} a^{8} x^{8}+4 e^{2 \mathit {atan} \left (a x \right )} a^{6} x^{6}+6 e^{2 \mathit {atan} \left (a x \right )} a^{4} x^{4}+4 e^{2 \mathit {atan} \left (a x \right )} a^{2} x^{2}+e^{2 \mathit {atan} \left (a x \right )}}d x \right ) a^{4} x^{2}-6 e^{2 \mathit {atan} \left (a x \right )} \left (\int \frac {x}{e^{2 \mathit {atan} \left (a x \right )} a^{8} x^{8}+4 e^{2 \mathit {atan} \left (a x \right )} a^{6} x^{6}+6 e^{2 \mathit {atan} \left (a x \right )} a^{4} x^{4}+4 e^{2 \mathit {atan} \left (a x \right )} a^{2} x^{2}+e^{2 \mathit {atan} \left (a x \right )}}d x \right ) a^{2}-1}{2 e^{2 \mathit {atan} \left (a x \right )} a \,c^{4} \left (a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1\right )} \] Input:

int(1/exp(2*atan(a*x))/(a^2*c*x^2+c)^4,x)
 

Output:

( - 6*e**(2*atan(a*x))*int(x/(e**(2*atan(a*x))*a**8*x**8 + 4*e**(2*atan(a* 
x))*a**6*x**6 + 6*e**(2*atan(a*x))*a**4*x**4 + 4*e**(2*atan(a*x))*a**2*x** 
2 + e**(2*atan(a*x))),x)*a**8*x**6 - 18*e**(2*atan(a*x))*int(x/(e**(2*atan 
(a*x))*a**8*x**8 + 4*e**(2*atan(a*x))*a**6*x**6 + 6*e**(2*atan(a*x))*a**4* 
x**4 + 4*e**(2*atan(a*x))*a**2*x**2 + e**(2*atan(a*x))),x)*a**6*x**4 - 18* 
e**(2*atan(a*x))*int(x/(e**(2*atan(a*x))*a**8*x**8 + 4*e**(2*atan(a*x))*a* 
*6*x**6 + 6*e**(2*atan(a*x))*a**4*x**4 + 4*e**(2*atan(a*x))*a**2*x**2 + e* 
*(2*atan(a*x))),x)*a**4*x**2 - 6*e**(2*atan(a*x))*int(x/(e**(2*atan(a*x))* 
a**8*x**8 + 4*e**(2*atan(a*x))*a**6*x**6 + 6*e**(2*atan(a*x))*a**4*x**4 + 
4*e**(2*atan(a*x))*a**2*x**2 + e**(2*atan(a*x))),x)*a**2 - 1)/(2*e**(2*ata 
n(a*x))*a*c**4*(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1))